
where
\(C_6 = \frac{3C_52^{2\mu _2c}c^2}{(2\mu _2c - 1)}\) , and
\( C_7 = \frac{3C_4^2 c^2 2^{4\mu _2 c}}{(\mu _2c)^2}\) , with \(C_4\)  and \(C_5\)  as defined in Lemma REF .

The batch size could be chosen as a function of the horizon \(n\) . Results in a similar spirit, i.e., where a stochastic gradient algorithm is run for \(n\)  iterations, and the parameters such as step-size and batch size are set as a function of \(n\)  are common in the literature, cf. [1]}, [2]}.

See Appendix REF .

The first term in (REF ) represents the initial error, and it is forgotten at a rate faster than \(O(1/n)\)  since \(\mu _2 c > 1/2\) .
The overall rate for the algorithm would depend on the choice of the batch size \(m\) , and it is apparent that the error \( E[\Vert \theta _{n} - \theta ^*\Vert ^2]\)  does not vanish with a constant batch size. As in the case of Theorem , we observe that the error \( E[\Vert \theta _{n} - \theta ^*\Vert ]\)  has an inverse dependence on the strong convexity parameter \(\mu _2\) .
